A topological space $X$ is homotopy dominated by a topological space $Y$ and denoted by $X\leqslant Y$ if there are maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ so that $g\circ f\simeq id_X$.

How can I find a finite polyhedron $P$ and a space $X$ so that $X\leqslant P$ but $P\leqslant X$ fails?

Thanks in advance.

Given an ordinal $\alpha$, I define $F_{n}(\alpha)$ as follows:

- $F_0(\alpha)=\alpha$
- $F_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the language of $\{\in\}$ has $(\mathrm{V}\models\phi(S,F_0(\alpha),F_1(\alpha)...F_{n}(\alpha)))\Leftrightarrow S=\beta$
- $F_\omega(\alpha)=\mathrm{sup}\{F_{n}(\alpha):n<\omega\}$

The existance of Reinhardt cardinals implies that $F_1(\alpha)$ exists for every sufficiently small $\alpha$. In fact, every Reinhardt cardinal is $F_1(\alpha)$ for some ordinal $\alpha$. (Yes, even though Reinhardt cardinals are not consistent with AC, they are all ordinals.)

However, this is the only information I could find on this. Could somebody else tell me what this is called if it is already named?

Is it inconsistent to say $F_n(\alpha)$ exists for $n\leq\omega$? Which $n$ is it inconsistent for?

Information gathered post-question:

- If there is a first-order $\phi$ such that $(\mathrm{V}\models\phi(S))\Leftrightarrow S=\alpha$, then $F_n(\alpha)=F_n(0)$ for every $n$. Here are some $\alpha$ for which this is true:
- The smallest Erdős initial ordinal, Mahlo cardinal, Ramsey, Rowbottom, Jonsonn, Inaccessible, Strongly Inaccessible, Limit, Weakly compact, ethereal, subtle (assuming each of these exists)
- Every other minimum initial ordinal for a first-order large cardinal axiom
- The second of each of those axioms, the third, the $\alpha$-th for $\alpha$ on this list (assuming they exist)
- Every $\alpha<\omega_1^{CK}$ (hint for proving this: Kleene's $\mathcal{O}$ is first-order definable)
- Every $\aleph_{\alpha}<\aleph_{\omega_1^{CK}}$ and $\beth_{\alpha}<\beth_{\omega_1^{CK}}$(a corrolary from above)

- $F_\alpha(\beta)$ is countable or $\omega_1$ (assuming ZF)

Honestly I think this is the first time $\beth_{\omega_1^{CK}}$ has been used in a theorem ever. If I'm wrong, please tell me.

In the 1988 Narosa edition of Ramanujan's *The Lost Notebook and Other Unpublished Papers*, on the first line of page 1 is the following:
$$ \Big(1+\frac1a\Big) \Bigg\{\frac{1}{(1-aq)(1-q/a)}+\frac{q(1+q)(1+q^2)}{(1-aq)(1-aq^3)(1-q/a)(1-q^3/a)}+\frac{q^2(1+q)..(1+q^4)}{..}+..\Bigg\} .$$
The infinite sum in braces with $a=-1$ has a $q$-series expansion$$ A(q) := 1 - q + 3q^2 - 2q^3 + 3q^4 - 3q^5 + 4q^6 - 3q^7 + 6q^8 - 4q^9 +\dots$$
which is the generating function of OEIS sequence
A292511 and I conjecture that $A292511(n-1) = -(-1)^n A260195(n)$ where $A260195(n)$ is the number of integer triples $[x,y,z]$ such that $1\le\min(x,z),\max(x,z)\le y$ and $y^2-(x^2-x+z^2-z)/2=n$.

Is this conjecture true?

It may help that G. E. Andrews and B. C. Berndt in *Ramanujan's lost notebook*, Part I, page 277, in discussing Ramanujan's equation (Entry 12.4.5), have a result in equation (12.4.23) that leads to
$$ q A(q) = \prod_{n>0} \frac{1+q^n}{1-q^n}
\sum_{n>0} -(-1)^n q^{n^2}\frac{1-q^{2n-1}}{(1+q^{2n-1})^2} $$
which gives an alternate way of getting the $q$-series expansion.

Which high-dimensional lattices (particularly $Z_n^*,D_n,D_n^*,A_n,A_n^*$), exhibit the following property shown in the attached diagram? Two 2D lattices are shown, with the lattice points in red, the Voronoi cells in black, and the origin cell in blue. The cells closest to the origin cell are shown in green. Shown in purple is a point that belongs to the origin cell, and is displaced a small distance to a different cell.

In the hexagonal lattice, we see that such a small displacement will only transport the purple dot to one of the closest neighboring cells (green). But in the cubic lattice, a small displacement can move the purple dot to a Voronoi cell which is not one of the closest neighbors. Which other high-dimensional lattices share this property with the cubic lattice? I have consulted Conway and Sloan's book, but this question did not appear to be addressed. If anyone knows the answer or can point me to helpful references, I'd really appreciate it.

I am simulating two objects on a grid, and checking how long they can run until the two objects meet. The two objects move randomly, and choose any block (up, down, left, right) randomly. If the side the objects randomly selects does not have a block, then the objects goes the other way. I'm generating the probability of the two objects *not* meeting after some time $t$.

This is a Markov process, and the states are $[1..n*n]$, where $n*n$ is the grid coordinate $(n,n)$. For grid of size two and three, here are the tables:

$\mathbf{Mat} = \begin{bmatrix}0&0.5&0.5&0\\0.5&0&0&0.5\\0.5&0&0&0.5\\0&0.5&0.5&0\end{bmatrix}$

$\mathbf{Mat} = \begin{bmatrix} 0 & 1/2 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 \\ 1/4 & 0 & 1/4 & 0 & 1/2 & 0 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 \\ 1/4 & 0 & 0 & 0 & 1/2 & 0 & 1/4 & 0 & 0 \\ 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 & 1/4 & 0 \\ 0 & 0 & 1/4 & 0 & 1/2 & 0 & 0 & 0 & 1/4 \\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 \\ 0 & 0 & 0 & 0 & 1/2 & 0 & 1/4 & 0 & 1/4 \\ 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 1/2 & 0 \end{bmatrix}$

So if an object is initially at $(0,0)$, then it's probability at $t$ of being at any position in the grid can be gotten from $$\mathbf{u} = [1,0,0 \:..\:0].(\mathbf{Mat})^{t}$$

Similarly for the second object, it would be $$\mathbf{v} =[0,0 \:..\:0, 1].(\mathbf{Mat})^{t}$$

So the probability of the two objects not meeting by some time $t$ would be $$P(t)= (1 - \mathbf{u}.\mathbf{v}) * P(t-1)$$

$P(t)$ certainly decreases with time $t$. But is this decrease random for each grid size? Like for grid size 2, $P(t)$ decreases as $[1,0.5,0.25...]$, which exponentially decreases. But for higher sizes, does $P(t)$ vary the same way? Thanks for any suggestion!

Everyone knows that the real line $\langle\mathbb{R},<\rangle$ is the unique endless complete dense linear order with a countable dense set. Suslin's hypothesis is the question whether we can replace separability in this characterization with the assertion that the order has the countable chain condition, that is, that every set of disjoint intervals is countable. In other words, Suslin's hypothesis, in the original formulation, is the assertion that the real line is the unique endless complete dense linear order with the countable chain condition.

**Question.** Does ZF plus the axiom of determinacy AD imply the original Suslin hypothesis?

Set theorists proved in ZFC that Suslin's hypothesis is equivalent to the assertion that there is no Suslin tree, which is a tree of height $\omega_1$ with no uncountable chains or antichains. And ZF plus the axiom of determinacy AD settles this version of SH by proving that $\omega_1$ is measurable and hence weakly compact and hence has the tree property and so under AD there is there is no Suslin tree.

So under AD there is no $\omega_1$-Suslin tree. But does this mean that there is no Suslin line?

The point is that refuting a Suslin tree does not seem directly to refute all Suslin lines in the non-AC context, and therefore it does not seem to settle the original Suslin problem under AD. So the question is: does ZF+AD settle the original Suslin problem?

Please feel free to post an answer explaining the precise details of the argument that AD implies there is no $\omega_1$-Suslin tree.

I heard this question this evening at a party at a conference in honor of Simon Thomas from a certain prominent set theorist, aged Scotch in hand, who told me that he would rather not be mentioned, but who said he was fine for the question to be posted.

Let me update the question (after Asaf's very nice answer) to ask about the situation where we also have the axiom of dependent choice DC. And perhaps one really wants to know about the case in $L(\mathbb{R})$ under AD.

**Question.** Does ZF+AD+DC imply the original Suslin hypothesis?

**Question.** Does $L(\mathbb{R})$ have a Suslin line assuming AD?

Using a summation method of divergent series such that $$S_1=1+2+3+\cdots=-\frac{1}{12},$$ find the sum of $$S_2=1-1+2-2+3-3+\cdots,$$ $$S_3=1-1+1+2-2+2+3-3+3+\cdots$$ etc. ($S_n$ for all $n$).

To be edited.

Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let

\begin{equation} d(T) = \sup \left \{ \frac{||x||_X}{||Tx||_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||_{\mathrm{op}} \end{equation}

Let \begin{equation} c(X,Y) = \inf \Bigl\{ d(T): T \mbox{ is an injective linear map from } X \mbox{ to }Y \Bigr \}. \end{equation}

Let $\ell^p_n$ be $\mathbb{R}^n$ with the $p$-norm and let $\mathcal{H}$ be an infinite dimensional Hilbert space. I would like to know if $c(\ell^1_n,\mathcal{H}) = O((\log n)^k)$ for some $k$. Alternatively, is it the case that $c(\ell^p_n,\mathcal{H}) = O_p((\log n)^{k(p)})$ for every $p > 1$?

There is a lot of literature about the Banach-Mazur distance but I have been unable to find information about linear embeddings (rather than isomorphisms). On the other hand, there is a well known result of Bourgain which asserts that a metric space with $n$ points can be embedded in $\mathcal{H}$ with distortion $O(\log n)$, but I don't know if this can be done in a uniform linear way.

I can't remember, what is the resulting unit after taking cos(10°)?

Ex: cos(10°) = 0.9885...________ what is the unit of the 0.9885?

The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk.

What is the analogue of this, over positive characteristic? Specifically, what replaces the Maurer-Cartan equation $$dx + \frac{1}{2}[x,x] =0$$ in positive characteristic?

Do formal groups enter in an essential way, as a substitute for Lie algebras, in deformation theory in positive characteristic? This point is not answered in the responses to a similar question Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?) .

For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $GL(n/2,\mathbb{C})$ and $Sp(n)$ structure on $M$, respectively. With additional integrability (the vanishing of the Nijenhuis tensor or the closedness of the almost symplectic form) one has a complex or symplectic structure on $M$.

Since CR and Contact geometry are the odd-dimensional cousins of the above geometries, it seems as though it would make sense for CR and Contact structures to come from a reduction of the structure group of the manifold. However, these structures tend to be defined more in terms of their integrability conditions. I suppose what I am really hoping for is a more unified theory among the different geometric structures. The Riemannian, Almost Symplectic, and Almost Complex structures share an intimacy with their "2 out of 3" property arising from properties of the intersections of the reduced structure groups, and it would be nice if the relationship between Almost Contact and Almost CR structures stemmed from a similar place, and furthermore that their integrability conditions corresponded to some sort of "flatness" on the $G$-structure.

So I ask, do these $G$-structure correspondences exist for the above geometries, and do they share an analogous relationship with their even-dimensional counterparts?

When one is checking whether two categorical variables are independent, it is customary to construct a contingency table and use the chi-square test. However, why can the chi-square distribution with k degrees of freedom (i.e. the sum of squares of k standard normal variables) stand proxy for contingency tables with k degrees of freedom? Is there some kind of proof that they are related? As far as I can see, most statistics textbooks just leave this unexplained.

I'm reading a research article lately, and got confused about a question. So, the fundamental theorem of Kruskal and Katona states that if each set in a given set system $\mathcal{A}$ has $k$ elements and $|\mathcal{A}|=m$, then the lower shadow of $\mathcal{A}$ is at least as large as the lower shadow of the initial segment of length $m$ of $N^{(k)}$ in the colex order. Here, $N^{(k)}$ is the family of all $k$-element sets of all the natural numbers. Then, the article mentioned that, as an easy consequence of this Kruskal-Katona theorem, we have the following lemma:

If $\mathcal{F}$ is a down-set, $||\mathcal{F}|| \leq || \mathcal{I}(|\mathcal{F}|) ||$. Here, $\mathcal{I}(m)$ is the initial segment of length $m$ of $N^{(<\inf)}$ in the colex order. Here, $N^{(<\inf)}$ is the family of all the finite subsets of all natural numbers. I did not see the reasoning behind this. Could anyone give me a hint of this? Many thanks for your time and attention.

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map.

Can $T$ be extended to a `positively linear map' on $L_0(M)_+$, the positive part of the *-algebra of $\tau$-measurable operators affiliated with $M$?

The idea is obvious. We express a positive element from $L_0(M)$ as the limit of an increasing net of positive elements $(x_a)$ in $M$. In this case we ought to define

$Tx = \sup Tx_a$,

as the net $(Tx_a)$ is increasing due to positivity of $T$. This definition readily works for $T=1_M\cdot \tau$, however in full generality there is an issue with the choice of $(x_a)$. I don't think it is obvious that $Tx$ does not depend on $(x_a)$.

It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled.

Here I'd like the "local models" to be something more enlightening than just "all fibrations". This will be impossible if one doesn't restrict to paracompact bases; I'd be happy even if the base is restricted to be a CW complex. For that matter, I'd be happy if it could be done over a manifold!

**Definition.**

A

is a pair $(\mathcal{U},\mathcal{F})$ where**class of local models**$\mathcal{U}$ is a subcategory of spaces, closed under intersection of open embeddings (in the examples here $\mathcal U$ is just all of $\mathsf{Top}$, but it seems natural to include it as a parameter)

For each $U \in \mathcal{U}$ we have a subcategory $\mathcal{F}_U \subseteq \mathsf{Top}/U$ of all maps over $U$, stable under pullback along open embeddings.

A map $p: E \to B$ is

if $B$ admits an open cover $B = \cup_i U_i$ such that**locally in $(\mathcal U, \mathcal F)$**each $U_i \in \mathcal U$

each pullback $E|_{U_i} \to U_i$ is in $\mathcal{F}_{U_i}$ (up to homeomorphism)

each transition map $E|_{U_i}|_{U_i \cap U_j} \to E|_{U_j}|_{U_i \cap U_j}$ is in $\mathcal{F}_{U_i \cap U_j}$.

**Remark.** Of course, really "being locally in $(\mathcal U, \mathcal{F})$" is a structure and not a property as suggested by the terminology -- $E \to B$ could be locally in $(\mathcal U, \mathcal{F})$ in many different ways (so the above definition really should be neatened up a bit -- I hope it's clear how to do this). With this in mind, it is natural to generalize the definition, replacing $(\mathcal U, \mathcal{F})$ by an arbitrary fibration over $\mathsf{Top}$ equipped with a map of fibrations to the codomain fibration $\mathsf{Top} \downarrow \mathsf{Top} \to \mathsf{Top}$. But I think this extra generality is not needed for the following.

**Examples.**

A map $p: E \to B$ is a fiber bundle iff it is locally in $(\mathsf{Top}, \mathsf{Triv})$ where $\mathsf{Triv}_U$ is the category of projections $F \times U \to U$ and homeomorphisms between them.

A map $p: E \to B$ is a fiber bundle with structure group $G$ iff it is locally in $(\mathsf{Top}, \mathsf{Triv}_G)$ where $(\mathsf{Triv}_G)_U$ consists of projection maps $F \times U \to U$ where $F$ is one among a specific list of $G$-spaces, and $G$-equivariant maps.

[Hurewicz and Huebsch] (

**Key example!**) If $B$ is paracompact, a map $p: E \to B$ is a fibration iff it is locally a fibration, i.e. locally in the class $(\mathsf{Top}, \mathsf{Fib})$, where $\mathsf{Fib}_U$ is the class of all fibrations over $U$, and homeomorphisms between them.(Less important) A smooth manifold is an identity map $B = B$ which is locally modeled on $\mathcal{U}$ being open subsets of Euclidean space and smooth maps, and $\mathcal{F}_U$ consisting of identity maps.

**Question.** Is there a class $(\mathcal U, \mathcal{F})$ *smaller* than $(\mathsf{Top}, \mathsf{Fib})$ (Example 3) such that for paracompact $B$, a map $p: E \to B$ is a fibration iff it is locally in $(\mathcal U, \mathcal{F})$?

I'd be happy with an answer that only works with further conditions on the topology of the fiber or the base.

Of course, one could take $\mathcal{U}$ to be an arbitrary collection of spaces such that every point in every space has a neighborhood contained in $\mathcal{U}$, with $\mathcal{F}$ being all fibrations over such spaces. This is not very enlightening *per se*, but e.g. if there's a particular choice of such $\mathcal{U}$ that sheds some light on the situation, that would be interesting.

I also haven't been specific about what kind of fibration I'm talking about. I think I'd be happy with any version -- Hurewicz, Serre -- even quasifibrations would be interesting.

**Analogy to algebraic geometry.** If we replace $\mathsf{Top}$ by the category of schemes (or even by functors $\mathsf{CRing} \to \mathsf{Set}$), then a quasicoherent sheaf on a scheme is modeled locally on modules over affine schemes. And a vector bundle is just a locally free coherent sheaf. So in algebraic geometry, there's something interesting to say about the local structure even of non-locally-trivial quasicoherent sheaves. I'm wondering if in topology, there's likewise something interesting to say about non-locally-trivial fibrations.

The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:

The minimal faithful representation has dimension $196883 = 47.59.71$

The Monster group can be realised as a Galois group $Gal$ $L(71)/{\mathbb{Q}(\sqrt{-71})}$ where $L(71)$ is a suitable field.

The appearance of $71$ in the above two cases (and possibly others) is not very surprising and might be reasoned out.

But the appearance of $71$ in a different area seems very intriguing:

The Monster is intimately connected to a special class of conformal field theories. These are the meromorphic $c = 24$ CFTs. The Monster here arises as the discrete automorphism group of the vertex operator algebra of one of the c =24 CFTs.

Schellekens in 1992 enumerated such CFTs and he found $71$ such CFTs! All these CFTs have a partition function of the form $$ Z(\tau) = j(\tau) + \mathcal{N} $$ where $j$ is modular invariant and $\mathcal{N} \geq -744$ is an integer. But any value of $\mathcal{N}$ won't work. Schellekens found $71$ values of $\mathcal{N}$ which will work.

Unfortunately, it is still not clear if the enumeration Schellekens made is exhaustive, i.e. if there are only exactly $71$ such theories.

Is the appearance of $71$ here just a coincidence? Or is it again connected to the Monster? It is hard to believe that this is just a coincidence.

Let $X\rightarrow Y\rightarrow Z$ be a stein factorisation. If we know the the fibres of the composite morphism is connected, then wouldn't it imply that $Y=Z$?

All spaces above are integral varieties.

If $X$ and $Y$ are two sets of $n$ independent, uniformly sampled points in the unit square, then standard methods can show that the expected minimum distance between points in $X$ and $Y$ is proportional to $1/n$, that is, $$E(\min_{i,j}\|x_i - y_j \|)\sim 1/n$$as $n\to\infty$. Is there anything similar that can be said when we have three sets of points $X,Y,Z$, and we look for the triangle whose perimeter is shortest? I.e., $$E(\min_{i,j,k}\|x_i-y_j\|+\|y_j-z_k\| + \|z_k - x_i\|)$$?

Let V=W[g], where g is P-generic over W for some poset P in W. Let F be a V-extender with critical point κ such that P ∈ VκW. If the support of F is sufficiently closed, say strength(F)=length(F)=λ for some inaccessible cardinal λ>κ, then F ∩W ∈ W. To the best of my knowledge, this is due do Hamkins-Woodin; the argument for it which I have in mind is the one written up in https://ivv5hpp.uni-muenster.de/u/rds/VM2_4.pdf .

Can this be proven in more generality? I.e., what if we don't assume ult(V;F) to be Card(P)-closed, is it still true that F ∩W ∈ W? Or is there a counterexample?

And can one even drop the hypothesis that F be a **total** V-extender, i.e., that it acts on some transitive model contained in V rather than on all of V?

Arakelov constructed a nice intersection theory on arithmetic surfaces. A key point is the notion of Green function for a Riemann surface, which will be involved in the ''part at infinity'' of the calculation of the intersection number between two Arakelov divisors.

Lately I've been reading the paper "P. Hriljac - Heights and Arakelov's intersection theory" and he defines the intersection theory on arithmetic surfaces by using just the notion of Neron function (on a Riemann surface). A Green function is in particular a Neron function and in the paper you may find the following comment:

The existence of Neron functions on curves over $\mathbb C$ (...) may be viewed as stemming from Arakelov, where he uses Green's functions in lieu of Neron functions. The Green's functions can be viewed as explicit realizations of Neron functions (...) Observe that there is no unique such family, which corresponds to choices of metrics as in [Arakelov main paper]. Among all Green's functions, there is one which can be selected to be "better" than all others (...)

Basically I don't understand what is the author saying with the above lines. In particular I don't understand why we choose Green functions as the "right" Neron functions involved in intersection theory.

This question may be relevant.